Use the ratio test to determine if $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\frac{5}{4}\right)^{2 n}(n!) $$ converges or diverges and justify your answer. Answer Attempt 1 out of 2 Using the ratio test, $$ \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\square $$. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\frac{5}{4}\right)^{2 n}(n!) $$

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We start with the series

$$
\sum_{n=1}^{\infty} (-1)^n \left(\frac{5}{4}\right)^{2n}(n!)
$$

Let

$$
a_n = (-1)^n \left(\frac{5}{4}\right)^{2n}(n!)
$$

Since the ratio test involves the limit of

$$
L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|,
$$

we first compute $$ a_{n+1} $$:

$$
a_{n+1} = (-1)^{n+1} \left(\frac{5}{4}\right)^{2(n+1)} ((n+1)!)
$$

Then the ratio is

$$
\left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{(-1)^{n+1} \left(\frac{5}{4}\right)^{2(n+1)} ((n+1)!)}{(-1)^n \left(\frac{5}{4}\right)^{2n}(n!)}\right|.
$$

Cancel the alternating sign factor:

$$
\left|\frac{(-1)^{n+1}}{(-1)^n}\right| = 1.
$$

Next, simplify the powers of $$\frac{5}{4}$$:

$$
\frac{\left(\frac{5}{4}\right)^{2(n+1)}}{\left(\frac{5}{4}\right)^{2n}} = \left(\frac{5}{4}\right)^2.
$$

Now, simplify the factorials:

$$
\frac{(n+1)!}{n!} = n+1.
$$

Thus, we have

$$
\left|\frac{a_{n+1}}{a_n}\right| = \left(\frac{5}{4}\right)^2 (n+1).
$$

We now compute the limit:

$$
L = \lim_{n\to\infty} \left(\frac{5}{4}\right)^2 (n+1).
$$

Since $$\left(\frac{5}{4}\right)^2$$ is a constant and $$ n+1 $$ tends to infinity, we have

$$
L = \infty.
$$

According to the ratio test, if $$ L > 1 $$ (in particular, if $$ L = \infty $$), then the series diverges.

Thus, the series

$$
\sum_{n=1}^{\infty} (-1)^n \left(\frac{5}{4}\right)^{2n}(n!)
$$

diverges.
The series $$ \sum_{n=1}^{\infty} (-1)^n \left(\frac{5}{4}\right)^{2n}(n!) $$ diverges because the limit of the ratio $$ \left|\frac{a_{n+1}}{a_n}\right| $$ as $$ n $$ approaches infinity is infinity, which is greater than 1.

Use the ratio test to determine if converges or
diverges and justify your answer.
Answer Attempt 1 out of 2
Using the ratio test, .

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Use the ratio test to determine if converges or diverges and justify your answer. Answer Attempt 1 out of 2 Using the ratio test, .